Exact radiation conditions on spheroidal boundaries with sparse iterative methods for efficient computation of exterior structural acoustics

Abstract

Exact radiation boundary conditions are derived from harmonic expansions in spheroidal coordinates and formulated in a finite-element method for the Helmholtz equation in unbounded domains. The use of spheroidal boundaries enables the efficient solution of scattering from elongated objects. Extending the ideas of Malhotra for the exact Dirichlet-to-Neumann (DtN) map defined on a sphere, a matrix-free implementation of the nonlocal DtN map for spheroidal boundaries, suitable for iterative solution of the resulting complex–symmetric system, is described. Efficient SSOR preconditioners together with Eisenstat’s trick based on the sparse matrix partition associated with the interior mesh and local part of the radiation boundary operator are used to accelerate convergence of Krylov-subspace iterative solution methods. Numerical examples are computed to demonstrate the efficiency and accuracy of the boundary treatments for high-frequency scattering from elongated structures. [Work supported by NSF PECASE.]